Real Analysis

A.Y. 2020/2021
Overall hours
Learning objectives
The course aims at introducing the students to some fundamental aspects of Real Analysis, with particular reference to Lebesgue and Hilbert spaces.
Expected learning outcomes
Students will acquire familiarity with the basic properties of the real analysis, with particular focus on the theory of Lebesgue and Hilbert spaces. Students should be able to independently produce proofs of elementary statements (aided also by the experience obtained through homework assignments), and be able to explain rigorously the theoretical knowledge and computational aspects learned from the lectures and assigned problems. Moreover they will improve their skills to work in small groups of fellow students.
Course syllabus and organization

Single session

Lesson period
First semester
Videos with lessons and exercises collections will be made available at the official web space of this course and they will cover the topic of each week. Live meetings during the classes hours may be scheduled by using The details of these meetings will be available in the course web page of the Ariel platform, as well as the videos and every kind of material needed for the course. The exams will be done by following the ways suggested on the web page of the university.
Course syllabus
1. Differentiation and integration: Review of the Lebegue integral. Integral functions and the Lebesgue Differentiation Theorem. Signed measures and the Radon-Nikodym Theorem. Differentiation of monotone functions. Functions of bounded variation, Absolutely continuous functions. Convex functions and Jensen's inequality.

2. Lp Spaces: Definition, Hölder and Minkowski inequalities, convergence and completeness. Comparison of notions of convergence. The dual space of Lp and the Riesz Representation theorem for Lp. Convolution and the inequalities of Young. Approximation in Lp by regular functions.

3. Hilbert spaces: Definition and fundamental properties. The Projection Theorem. Orthonormal bases. Continuous linear functionals and the Riesz Representation Theorem for Hilbert Spaces. Bilinear forms and the Lax-Milgram Theorem. Orthonormal bases and separability. Expansions in Fourier series and fundamental examples of complete orthonormal systems. Convolutions kernels and pointwise convergence of Fourier series. Bounded linear operators. Self-adjoint operators and compact operators. Spectral Theorem for compact self-adjoint operators.
Prerequisites for admission
Knowledge of measure theory and integration including the theory of Lebesgue and Hausdorff. this material is covered in Mathematical Analysis 4.
Teaching methods
In order to facilitate the learning process, classroom lectures and problem sessions will be offered to the students, who are strongly encouraged to participate in these activities.
Teaching Resources
- R.L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Monographs and Textbooks in Pure and Applied Mathematics, Vol 43, Marcel Dekker, Inc., New York, 1977.
- G. B. Folland, Real Analysis, Modern Techniques and Their Applications, John Wiley & Sons, Inc, New York, 1999.
- E. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration and Hilbert Spaces, Princeton Lectures in Analysis, Vol. III, Princeton University Press, Princeton, 2005.
- E. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton Lectures in Analysis, Vol. I, Princeton University Press, Princeton, 2003.
Assessment methods and Criteria
The final examination consists homework assignments (tipically three assignments with various exercises) and an oral exam.

- In the homework assignments, the student must solve some problems in the form open-ended questions, with the aim of assessing the student's ability to solve problems regarding the main results and definitions presented in the course.
- The oral exam can be taken only if the homework assignments have been performed in a regular manner. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding the main results and definitions presented in the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The final examination is passed if the oral exam is successfully passed. Final marks are given using the numerical range 18-30, and will be communicated immediately after the oral examination.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 9
Practicals: 24 hours
Lessons: 49 hours
My office: Dipartimento di Matematica, via Saldini 50, first floor